From the research I did for the 1950s article on Cal11: (http://www.calibre11.com/history-of-tag-heuer-1950s/ ), the connection started in the early '50s with the US jeweller Zales taking several Heuer watches and co-branding them with their watch arm, Baylor. Non-chrono watches was a throw of the dice for Heuer, but it just made them a smaller fish in a bigger pond so when Jack joined the firm in 1958, it was something he wanted to change, to fight it out again with just a few Swiss manufacturers of chronographs. Hard to tell exactly when now, but the remaining stock (which did include some alarm watches) was remaindered to Zales in around 1959 and they sold them themselves. Heuer carried on with time only watches for a year or two, just into the '60s but then canned it to concentrate on chronos (for the best part of a couple of decades).

Worth remembering these were all Heuer watches with Baylor branding, not the reverse, and manufacturing stopped around the very end of the '50s. I don't know exactly when the watch being referred to was made, but it does seem a bit late to fit in with the above. We'd need to see it up close in the metal to make a call on the Heuer shield.